Risky Equity Valuation

Notes using the binomial option pricing model

Imagine this

You are considering to found an equity fund to track risky investments in privately owned organizations

• You have a history of stock prices
• You know that today’s stock price is the present value of stockholders’ free cashflow for the foreseeable and not-so-foreseeable future
• You need a back of the envelope way to describe how stock prices and value evolve
• You also want to use this evolution to understand how a claim contingent on stock prices also evolves
• And the present value of such a claim

What do you want to know?

• Is there a distribution you can use to describe the evolution and also price a claim on the stock prices?

Why?

• So you can acquire or divest an asset, a project, or anything that generates cash flows over time and has the same risk profile as the stock price.

What’s a binomial?

Suppose we know that the current stock price of Ross Stores (NASDQ: ROST) \(S\) is $109 per share. We might consider a simple forecast of the uncertain level of the asset’s value in one day, or even several days into the future: it might go up or it might go down. That’s a binomial.

We picture the binomial view of an anticipated stock price \(S\) as branches from a root of \(S_0=\$109\) per share today at time \(t=0\). How can we forecast asset value at the end of one day \(S_1\)? We can do so by supposing that the stock price might simply rise or fall. Both up \(u\) and down \(d\) stock price outcomes, \(S_{1,u}\) and \(S_{1,d}\), might occur in \(t=1\) day.

We might also want to represent our optimisim or pessimism about how often up moves and down moves might occur. Let’s assume just for a minute that we are optimistic about the future so that the probability of any up move in this stock is \(p = 0.60\). If the probability of an up move is 0.60, then the probability of a down move must be

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Probability of a down movement is 1 minue the probability of an up movement or \(1-p=1-0.06=0.40\).

We can hover over the diagram below and locate the initial stock price and the up and down possibilities as well as probabilities of an up or down movement.

Now we can suppose that the stock price grows or declines by an amount. That amount can be learned from the sample standard deviation \(\sigma\) of the rate of return of the stock price. Rates of return are just growth (or decline) rates. So that the one day ahead stock price will go up to

\[ S_{1,u} = S_0 + S_0 \sigma = S_0(1+\sigma) \approx S_0 e^{\sigma} \] Where we use the approximmation \((1+\sigma) \approx e^{\sigma}\). We must keep in mind that this is the one day standard deviation.

Let’s get this a little more down to earth. We find that over the past 251 trading days that the stock return daily standard deviation is 0.0165 or 1.65% per day. If we multiply this by 251 days a $1 invested today would grow into 62.8970964. This means that after 251 up jumps from today we would get this amount. Intuitively we might feel this is not very likely! Anyway, after just one up jump (a stock price move for one day) and if today’s stock price is $109 per share, then

\[ S_{1,u} = S_0 + S_0 \sigma = S_0(1+\sigma) \approx S_0 e^{\sigma} = 109 e^{0.0165} = 110.8134196 \]

What goes up might just as well go down. If stock returns have a positive \(\sigma = 0.0165\), and since \(\sigma = var^{1/2}\), then it is possible for a negative or down turn in the return. Now the factor is $e^{-0.0165}=0.9836354, a discount to the current stock price of $109 per share.

\[ S_{1,d} = S_0 - S_0 \sigma = S_0(1-\sigma) \approx S_0 e^{-\sigma} = 109 e^{-0.0165} = 107.2162564 \]

If the stock could jump up 251 times it might decline that many times too. In that scenario, the stock price would move from $109 to what level?

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1.7329894

What a decline!

What would be the expected value of the stock price in one day? This question is just a weighted average of the probabilities of up and down with the stock price outcomes, a random variable in one day.

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\[ E(S_1) = p S_{1,u} + (1-p) S_{1,d} = (0.60)(110.81)+(0.40)(107.2163)=109.37 \]

Just a tad above today’s price.

Then there were two

Yes, two draws of stock prices, one after the other, one conditional on the other. Let’s look at this up and down tree.

SOme description is in order. At the top is right now, day 0. The second row is day 1. The third row is day 2. If the stock goes up from day 0 to day 1, then the value is \(S(1,u)\). Similarly if the stock goes down. If after the stock goes up at day 1, then the stock can either go up to \(S(2,uu)\) (that is, jump up twice) to go down to \(S(2,ud)\) (that is, after jumping up it then jumps down). The same thoughts will occur when the stock goes down to \(S(1,d)\).

How many paths are there to get to the third row where we have forecasted the 2nd day’s stock price?

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For \(S(2,uu)\) there is only one path, up and up.

For \(S(2,ud=du)\) there are two paths, up and down, and down and up. It does not matter, they both lead to the same node.

For \(S(2,dd)\) there is only one path, down and down.

How often?

We can compute outcomes all day. But how often does a node occur? We remember that our task is to forecast stock prices in the future. To do that we realize that stock prices occur in a probable range. That means they are random variables, where the adjective random has the notion of an indiscriminate sampling of prices. But a random variable is not so colloquially indiscriminate as to not have a notion of a frequency of occurrence. The relative frequency, as we continue to see, is what we measure to be probability. Allowing probability into our lives also admits our beliefs into the analysis.

So what is the probability of a stock price after one up and one down jump?

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This is an example of a both-and event. Both an up and a down jump must occur. The probability of an up jump, according to our analysis is 0.60. The probabiilty of a down jump is then just \(1-0.60=0.40\). Intuitively, if stock prices were to rise, that would account for 60% of our experiments. Then after an up jump, the stock falls, so out of 60 pf a humdred experiments with stock prices, 40% of those might occur.

Thus the probability of an up jump followed by a down jump is

\[ Prob(S(2, ud)) = p^1(1-p)^1 = (0.60)^1(0.40)^1 = 0.24 \]

THis is the probability of a single up and down combination.

Hang on! In the stock price tree we just admitted that there are two paths to get to a stock price that experiences both an up and a down jump. Okay, that would then mean there are two mutually exclusive ways of getting to an outcome like that. EIther the stock price jumps up then jumps down, or the stock price jumps down then up. This is thus an either-or event. We must therefore add the independently occurring probabilities.

\[ Prob(s(2, ud) \cup S(2, du)) = Prob(S(2, ud)) + Prob(S(2, du)) = 0.24 + 0.24 = 2(0.24) = 0.48 \]

Thus the complete answer is that 48% of the time we might experience a stock price that jump up one day and down another.

How would we calculate the \(S(2,du=ud)\) outcome?

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Let’s use the up and down sequence. We already know that

\[ S_{1,u} = S_0 + S_0 \sigma = S_0(1+\sigma) \approx S_0 e^{\sigma} = 109 e^{0.0165} = 110.8134196 \]

Yes, the stock price rises. After the rise to 110.8134196, the stock price now jumps down. We take our up estimate and condition it with a down movement like this

\[ S_{2,ud} = S_{1,u} - S_{1.u} \sigma = S_{1,u}(1-\sigma) \approx S_{1,u} e^{-\sigma} = 110.8134196 e^{-0.0165} = 109 \] Yes, right back where we started. Do we not observe a symmeterical outcome? Yes, indeed we do.

The \(S(2, du=ud)\) outcome occurs 48% of the time given our optimism in this market.

Ups and downs

To forecast the up and down movements of the stock price or any asset whose value is \(A\) in one year we can use the normal distribution, with a twist: the log normal random variable. Here’s what we will assume over a small interval of time \(\Delta t\):

1. Use small movements in the present value of assets \(A_t\) so that \(\Delta A_t = A_{t+\Delta t} - A_t\)
2. Let the small change in time, \(\Delta t\) get as small as you like
3. Approximate returns as the log value relative \(ln(v_{t+\Delta t} / v_{t}) \approx (v_{t+\Delta t} / v_{t}) - 1\), and this works well with small movements in value from \(t\) to \(t + \Delta t\)
4. In that small interval of time, \(\Delta t\) use a constant average rate of return \(\mu \Delta t\) and a constant variance of return \(\sigma^2 \Delta t\) which is the same as using a constant standard deviation of \(\sigma \sqrt{\Delta} t\)

Armed with all of these assumptions we see that from \(t=0\) to \(t\) we can accumulate value in small time increments \(\Delta t\) with this simulation:

\[ A_t = A_0 exp((\mu - 0.5 \sigma^2) \Delta t + \sigma \sqrt{\Delta t} \, z) \]

where, \(z ~ Normal(0,1)\), the standard normal (mean 0 and variance 1) distribution. Now for the lognormal bit. Divide both sides by \(S_0\) and take (natural) logarithms to get

\[ ln(A_{t+\Delta t}/A_t) = (\mu - 0.5 \sigma^2) \Delta t + \sigma \sqrt{\Delta} t \, z) \]

We thus are simulating asset returns across a time period of \(\Delta t = (t+\Delta t) - t\). For a given initial level of \(A_t\) at the beginning of the period, the logarithm of \(A_{t+\Delta t}\) is normally distributed. But not only that, we can bifurcate stock price movements at time \(t\) in the following way.

If we let up movement factors \(u\) be

\[ u = e^{\sigma \sqrt{\Delta t}} \] and down movement factors \(d\) be symmetrically the inverse of \(u\)

\[ d = e^{-\sigma \sqrt{\Delta t}} = \frac{1}{u} \] and along with the future value \(R\) of depositing a $1 bill into a bond that yields \(r\) per year in a holding period of \(\Delta t\)

\[ R = e^{r \Delta t} \] we have all of the pieces we need to forecast movements in \(S\), or whatever variable of our fancy, across whatever time periods we want, daily, monthly, quarterly, semiannually, annually.

For example supose we estimate \(\sigma = 0.40\) in one year (or per annum), and \(r = 0.05\) per annum. If we let \(t=1\) year and divide the year equally into \(n=4\) quarters, then we would compute

\[ \Delta t = \frac{t}{n} = \frac{1}{4} = 0.25 \]

This approach will allow us to calculate quarterly returns and up and down factors. If \(n=12\) we could break a per annum estimate into monthly estimates using \(\Delta t = 1/12 = 0.0833\). Using this convention we get

\[ u = e^{\sigma \sqrt{\Delta t}} = e^{0.40 \sqrt{0.25}}=1.22 \] \[ d = e^{-\sigma \sqrt{\Delta t}} = \frac{1}{u}=\frac{1}{1.22}=0.82 \] \[ R = e^{r \Delta t} = e^{0.05(0.25)} = 1.013 \] Now we can grow any stock price (or any asset present value for that matter) trees we might care to. Using our tree branches we have this forecast in general

and this in particular for one quarter of one year

Now for the pricing magic.

Very pricey indeed

Suppose we want to value the equity of an organization. We can view equity \(E\) as an American call option written on the value of the assets \(A\) (working capital plus fixed assets) of an organization with exercise value equal to the long-term debt of the organization. The payoff for equity at some future date \(t=T\) is \(max(E-D,0)\). Also suppose that debt \(D=35\).

Let’s use our forecast of \(A\) we worked out above for one quarter again. We can build a tree for Equity contingent on the Asset tree:

Now we build a portfolio of \(a\) units of the asset and \(b\) units of a riskless bond and find \(a\) and \(b\) such that this homemade portfolio’s payoff is the same as a the contingent claim called equity. At \(t=1\) we have two constraints, one for the up movement in the portfolio and one for the down movement in portfolio. Each $1 of the bonds grows into \(\$1.013\) by the end of a quarter.

\[ 48.80a+1.013b=13.80 \] \[ 32.80a+1.013b=0 \] We can solve these simultaneous equations for \(a=0.8625\) and \(b=-27.9270\). At time \(t=0\) the portfolio will equal the present value of equity \(E\).

\[ 40(0.8625)-27.9270 = 6.573 \] This is the market value of equity. Book value would be \(40-35 = 5\) also known as intrinsic value. Thus the market to book ratio is \(6.5730/5 = 1.314\). The amount of market value in excess of book value \(6.570-5=1.570\) is called extrinsic value due to the volatility of the underlying asset \(A\). We have just “priced” equity.

In one more step we can get the risk neutral, market-clearing, probability of an up movement. This probability obeys the risk neutral solution for \(a\) and \(b\) we just found to value equity. The present expected value of equity \(E\) is in one period

\[ E = [pE_u + (1-p)E_d]e^{-r\Delta t} \] \[ 6.573 = \frac{p(13.80) + (1-p)(0)}{1.013} \] which yields \(p = 0.48\). We can use this probability to value any future up movement and still use the risk free rate to value expectations of forward value. In general if

\[ E = Aa + b = \frac{pE_u + (1-p)E_d}{R} \] we can show that

\[ p = \frac{R - d}{u - d}=\frac{1.013-0.82}{1.22-0.82}=0.48 \] We will use this relationship in the next section with cash flows, debt, decisions, and more than one period.

Multi-period valuation

Now for the icing on the cake. We have all of the pieces to value contingent claims like equity with multiple periods of cash flow and debt as well. We recall that equity is modeled as an American call option. Let’s extend our example from one quarter to four quarters. The asset tree \(A\) now looks like this

We can specify quarterly projected cash flows and debt levels next. The 4th quataer nodes are just eh American call option payoff at the end of the horizon or \(max(A-D,0)\). We work backwards from this last date to the third quarter. The American option decision is between early exercise with payoff \(72.88-20\) or keeping the option alive, receiving cash flow of \(5\) and the present expected value of waiting until quarter 4 to exercise the option. In this case, the rational thing to do is to keep the option alive.

\[ 58.14 = max\left(72.88-20, 5 + \frac{0.48(69.02) + (1-0.48)(39.67)}{1.013}\right) \]

With these calculations and decisions at each node in the lattice and the asset tree we can develop the following equity value tree.

Equity value is $40.75. Book value is only $20. Market value ends up being more than twice book in this scenario. Now to sensitize ourselves to changes in assumptions to complete this portion of the analysis.

Exercise

For this exercise we will access the following data:

• daily closing prices for up to five (5) years for a traded stock
• recent balance sheet, income statement, and statement of financial condition for the traded stock’s company
• the Excel spreadsheet equityvalue-1.xlsx.

Using this work flow we will compute and analyze the followng:

1. We transform balance sheet information in way: Invested capital (assets) = working capital (current assets - current liabilities) + fixed assets (total assets - working capital). This is the book value of invested capital. We will calculate the market value of invested capital later in the process below.

2. Next we calculate Long term debt (debt) = invested capital - book value of equity. Again later we will assumed that the book value of debt is a good approximation for the the market value of debt.

3. We calculate the natural logarithm of closing stock price relatives for each day starting with the observation at day 2 of the data set, that is, for day \(t\) and the previous day \(t-1\), daily return is

\[ r_t^{\,ticker} = ln\left(\frac{P_t}{P_{t-1}}\right) \]

4. Then we compute the sample daily mean and standard deviation of the \(r_t^{\,ticker}\) series. Then annualize the daily return, that is, transform the daily mean to an annual mean return basis by multiplying \(\Delta t = 252\) times \(r_t^{\,ticker}\) to the daily mean estimate. Similarly transform the daily estimate of the return standard deviation, \(r_{sd}\), by multiplying this estimate by \(\sqrt{\Delta t} = \sqrt{252}\).

\[ r_{mean}^{\,\,\,annual} = r_{mean}\Delta t = \left[\sum_{t=2}^T \left(\frac{r_t^{\,ticker}}{T}\right)\right]\Delta t \] \[ \sigma_{E} = r_{sd}\sqrt{\Delta t} = \left[\sum_{t=2}^T \left(\frac{(r_t^{\,ticker}-r_{mean})^2}{T-1}\right)\right]\sqrt{\Delta t} \] The mean and variance can be computed over the entire sample or as a rolling calculation to get a sense of the time varying properties of stock return movements.

5. Further transform the annualized equity return volatility estimate into an invested capital (asset) volatility estimate. First, we can model the present value of invested capital (\(V\)) as a portfolio of long term debt (\(D\)) and equity (\(E\)). We remember that the present value of invested capital is implied in the present values of debt and equity. We can observe the present values of debt and equity in the market place; we can’t observe the present value of invested capital. We infer that \[ V = D + E \] We estimate \(E\) from the current market capitalization of equity, that is, the current market (traded) price of shares of stock, \(p_E\), times the number of outstanding shares of stock (ex-dividend), \(N_E\). We will assume that the present value of debt, \(D\), is at current book value. This yields \[ V = D + p_E N_E \]
6. We can think of the variance of debt as 0 for this purpose and focus on folding in the variance of stock return \(\sigma_{E}^{2}\) on the assumption that there is no correlation between debt and equity returns and with \(\sigma_D=0\) we have.

\[ \sigma_{V}^2 = \left(\frac{D}{V}\right)^2\sigma_D^2 + \left(\frac{E}{V}\right)^2\sigma_E^2 \] \[ \sigma_{V} = \left(\frac{E}{V}\right)\sigma_E \] where, \[ E = p_E N_E \] and \[ V = D + E \] with \(D\) at book value.

7. Next we use \(\sigma_V\) to estimate asset return volatility, constant debt, a one year treasury rate, across 4 quarters in a year, with four quarters of estimated future cash flow for equity holders to decide to reinvest in the company (additions to retained earnings) or as dividends to build an invested capital binomial tree, up and down binomial factors, and risk neutral probabilities.

8. We then use the forecasted asset tree, free cash flows to shareholders, and the one year treasury rate to develop an equity valuation modeling equity as an American option.

9. Sensitize the equity valuation by varying input parameters including debt, asset volatility, initial asset value, and the treasury rate.

References

Brealey, R, Myers, S. and Allen, F. (2016). Principles of Corporate Finance. 12th ed. McGraw-Hill. Chapters

Haug, E.G. (2007). The Complete Guide to Option Pricing Formulas. 2nd ed. McGraw-Hill.

Hull, J.C. (2007). Options, Futures, and Other Derivatives. 7th ed. Pearson Education.